A Hankel norm for quadrature rules solving random linear dynamical systems

In linear time-invariant dynamical systems, random variables are included to quantify uncertainties. The solution can be expanded into a series with predetermined orthogonal basis functions, which depend on the random variables. We define a norm of Hankel-type associated to a truncated series. A quadrature rule or a sampling method yields approximations of the unknown time-dependent coefficient functions in the truncated series. We arrange a Hankel norm for the quadrature technique in this context. Assuming a convergent sequence of quadrature rules, we show that the Hankel norms of the quadrature methods converge to the Hankel norm of the truncated series. Hence a numerical method is obtained to calculate the Hankel norm of the truncated series approximately. Results of numerical computations confirm the convergence property in a test example.